Overview
The lecture explains how to analyze and graph rational functions, focusing on finding vertical and horizontal asymptotes, intercepts, symmetry, and plotting key points.
Identifying Asymptotes
- Vertical asymptotes occur where the denominator equals zero and cannot be canceled, making the function undefined at those x-values.
- For ( y = \frac{x-2}{x-3} ), the vertical asymptote is at ( x = 3 ), so the domain is all real numbers except ( x = 3 ).
- Horizontal asymptotes depend on the degrees of numerator and denominator:
- If degrees are equal, it's ( y = ) (leading coefficient of numerator) ÷ (leading coefficient of denominator).
- If numerator's degree is less, ( y = 0 ); if greater, no horizontal asymptote.
- In this example, both degrees are 1, so the horizontal asymptote is ( y = 1 ).
- Slant (oblique) asymptotes occur if the numerator's degree is one greater than the denominator's.
Finding Intercepts
- The x-intercept occurs where ( y = 0 ); set the numerator equal to zero and solve (( x = 2 ) here).
- The y-intercept is found by setting ( x = 0 ); substitute into the equation (( y = \frac{-2}{-3} = \frac{2}{3} )).
Symmetry and Function Type
- Test if a function is even or odd by plugging in ( -x ).
- If the function remains unchanged, it is even (symmetric to the y-axis).
- If all signs change, it is odd (symmetric to the origin).
- ( y = \frac{x-2}{x-3} ) is neither even nor odd.
Plotting Additional Points
- To better sketch the graph, calculate y-values for x-values around the asymptotes (e.g., ( x = -1, 4, 5 )).
- The graph may approach but will not cross vertical asymptotes; it can cross horizontal asymptotes.
Key Terms & Definitions
- Rational Function — A function expressed as one polynomial divided by another.
- Vertical Asymptote — A vertical line where the function becomes undefined (denominator = 0).
- Horizontal Asymptote — A horizontal line the graph approaches as ( x ) goes to ±infinity.
- Slant Asymptote — An oblique line approached by the function when numerator's degree is one greater than the denominator's.
- X-Intercept — Point(s) where the graph crosses the x-axis (( y = 0 )).
- Y-Intercept — Point where the graph crosses the y-axis (( x = 0 )).
- Even Function — Symmetric about the y-axis.
- Odd Function — Symmetric about the origin.
Action Items / Next Steps
- Watch the recommended video on slant asymptotes and full graphing examples.
- Prepare for the next lecture, which will cover rational functions with holes.